The vortex actually constitutes a classical solution in quantum field theory (QFT) [1]}. The vortex cannot be obtained from perturbative QFT as it is a non-perturbative solution. It turns out that since the size of the vortex is much larger than its compton wavelength, the classical non-perturbative solution constitutes a valid solution to the QFT (i.e. a very good first approximation) [1]}. Perturbation theory can then be used to obtain one-loop quantum corrections to the vortex by quantizing about the classical configuration. In particular, quantum fluctuations of the scalar field will change the nature of the potential as there will now be logarithmic terms besides the usual terms [3]}, [4]}. The critical exponent of \(1/2\)  will therefore change as a consequence of these quantum corrections. So an interesting and pertinent problem to solve for the future is to determine the critical exponent of the non-minimally coupled vortex in an AdS\(_3\)  background after quantum corrections. This would be a considerably more complicated calculation than say the quantization about the \(1+1\) -dimensional kink in Minkowski spacetime [1]} as we have one extra spatial dimension and a curved space background.
